Subgroups — #26116450

Order 12032.

684 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251740	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic
3	#20241268	C₂	1	4	yes	maximal
4	C₄₇×#1797	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic, complemented
6	C₂×#13094003	—	2	1	no	—
7	C₂²×C₄×Dic₄₇	—	2	1	no	—
8	#15689029	—	2	1	no	—
9	#15689021	—	2	1	no	—
10	#15689021	C₄	1	2	yes	complemented
11	#15688993	C₄	1	1	yes	characteristic, complemented
12	#15688983	—	2	1	no	—
13	#15688983	C₂²	1	2	yes	—
14	C₂×C₄₇⋊#95	—	2	1	no	—
15	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
18	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₄₇×#336	—	2	1	no	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂³×Dic₄₇	—	2	2	no	—
22	#13094024	—	4	2	no	—
23	#13094003	—	4	2	no	—
24	#13094003	—	4	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₄₇⋊C₄	—	2	2	no	—
31	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	2	4	no	—
33	C₂×C₄×Dic₄₇	—	4	2	no	—
34	C₂×C₄×Dic₄₇	—	4	2	no	—
35	C₄₇⋊#95	—	2	4	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₄₇⋊#95	—	2	2	no	—
38	C₄₇⋊#95	—	2	2	no	—
39	C₄₇⋊#95	Q₈	1	4	yes	complemented
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	—	4	2	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₄⋊C₈×C₄₇	—	4	2	no	—
50	C₂²⋊C₈×C₄₇	—	2	4	no	—
51	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	4	1	no	—
57	C₂²×C₄×C₄₇	—	4	1	no	—
58	C₂²×C₄×C₄₇	—	4	1	no	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	D₈	1	2	yes	—
64	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	—	4	2	no	—
75	C₂²×Dic₄₇	—	4	2	no	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	C₂²×Dic₄₇	—	4	2	no	—
78	C₂×Dic₉₄	—	4	2	no	—
79	C₂×Dic₉₄	—	4	2	no	—
80	C₂×Dic₉₄	—	4	2	no	—
81	C₂×C₈×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	—	4	2	no	—
83	C₂×C₈×C₄₇	—	4	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	4	1	no	—
87	C₄²×C₄₇	—	4	1	no	—
88	C₂²⋊Dic₄₇	—	4	2	no	—
89	C₂²⋊Dic₄₇	—	4	2	no	—
90	C₄⋊Dic₄₇	—	4	2	no	—
91	C₄⋊Dic₄₇	—	4	2	no	—
92	C₄⋊Dic₄₇	—	4	2	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₄⋊Dic₄₇	—	4	2	no	—
95	Dic₄₇⋊C₄	—	4	4	no	—
96	C₄×Dic₄₇	—	4	8	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	—	2	1	no	—
101	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
103	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂³×C₄₇	#137	1	2	yes	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	4	no	—
122	C₂×C₄×C₄₇	#99	1	2	yes	—
123	C₂×C₄×C₄₇	#102	1	2	yes	—
124	C₂×C₄×C₄₇	#102	1	2	yes	—
125	C₂×C₄×C₄₇	#104	1	2	yes	—
126	C₂×C₄×C₄₇	#104	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
130	C₂×Dic₄₇	—	2	8	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	Dic₉₄	—	8	4	no	—
144	C₈×C₄₇	—	8	2	no	—
145	#9909	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	#391	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
156	C₂²×C₄₇	#410	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
158	C₂²×C₄₇	#419	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
161	C₂²×C₄₇	#474	1	2	yes	—
162	C₂²×C₄₇	#475	1	2	yes	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	2	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	2	2	no	—
168	C₄×C₄₇	—	4	1	no	—
169	C₄×C₄₇	#327	1	4	yes	—
170	Dic₄₇	—	4	8	no	—
171	Dic₄₇	—	8	2	no	—
172	Dic₄₇	—	8	2	no	—
173	#2857	—	47	1	no	—
174	#1797	D₄₇	1	1	yes	characteristic
175	C₄×#95	—	47	1	no	—
176	#1294	—	47	4	no	—
177	C₂×C₄₇	—	2	2	no	—
178	C₂×C₄₇	—	2	2	no	—
179	C₂×C₄₇	C₂×#327	1	1	yes	characteristic
180	C₂×C₄₇	#1502	1	1	yes	characteristic
181	C₂×C₄₇	#1506	1	1	yes	characteristic
182	C₂×C₄₇	#1508	1	1	yes	characteristic
183	C₂×C₄₇	#1988	1	1	yes	characteristic
184	C₂×C₄₇	#1994	1	1	yes	characteristic
185	C₂×C₄₇	#1996	1	1	yes	characteristic
186	C₂×C₄⋊Q₈	—	47	1	no	—
187	C₂×C₂²⋊Q₈	—	94	1	no	—
188	C₂²×C₄²	—	94	1	no	—
189	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
190	C₂×C₄⋊C₈	—	2	1	no	—
191	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
192	#391	—	94	1	no	—
193	#389	—	47	2	no	—
194	#389	—	94	1	no	—
195	#384	—	47	1	no	—
196	#380	—	47	2	no	—
197	#380	—	94	1	no	—
198	C₂×#95	—	47	2	no	—
199	C₂×#95	—	94	1	no	—
200	#336	—	2	1	no	—
201	C₄₇	#9909	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	94	1	no	—
203	C₂³×C₄	—	94	2	no	—
204	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
205	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
206	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₈	—	2	2	no	—
208	C₄⋊Q₈	—	188	2	no	—
209	C₂²⋊Q₈	—	188	2	no	—
210	C₂²⋊Q₈	—	188	2	no	—
211	C₂×C₄⋊C₄	—	47	2	no	—
212	C₂×C₄⋊C₄	—	47	2	no	—
213	C₂×C₄⋊C₄	—	94	2	no	—
214	C₂×C₄⋊C₄	—	94	1	no	—
215	C₂×C₄⋊C₄	—	94	1	no	—
216	C₂×C₄⋊C₄	—	94	1	no	—
217	C₂×C₂²⋊C₄	—	94	2	no	—
218	C₂×C₄²	—	4	2	no	—
219	C₂×C₄²	—	94	4	no	—
220	C₂×C₄²	—	188	2	no	—
221	C₂×C₄²	—	188	2	no	—
222	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
223	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
226	C₄⋊C₈	—	4	2	no	—
227	C₂²⋊C₈	—	2	4	no	—
228	#95	—	47	4	no	—
229	#95	—	94	4	no	—
230	#95	—	94	2	no	—
231	#95	—	94	2	no	—
232	#95	—	94	2	no	—
233	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
234	C₂×Q₈	—	188	2	no	—
235	C₂×Q₈	—	188	2	no	—
236	C₂×Q₈	—	188	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	94	4	no	—
245	C₂²×C₄	—	94	4	no	—
246	C₂²×C₄	—	94	4	no	—
247	C₂²×C₄	—	94	2	no	—
248	C₂²×C₄	—	94	2	no	—
249	C₂²×C₄	—	188	2	no	—
250	C₂²×C₄	—	188	2	no	—
251	C₂²×C₄	—	188	2	no	—
252	C₂²×C₄	—	188	2	no	—
253	C₂²×C₄	#10625325	1	2	yes	—
254	C₂²×C₄	D₃₇₆	1	2	yes	—
255	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
257	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
258	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
260	C₂²×C₄	#10625352	1	2	yes	—
261	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
262	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₄⋊C₄	—	188	2	no	—
267	C₄⋊C₄	—	188	2	no	—
268	C₄⋊C₄	—	188	2	no	—
269	C₄⋊C₄	—	188	4	no	—
270	C₄⋊C₄	—	188	2	no	—
271	C₄⋊C₄	—	188	2	no	—
272	C₂²⋊C₄	—	188	2	no	—
273	C₂²⋊C₄	—	188	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	188	8	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	2	1	no	—
283	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
284	C₂³	#13094004	1	2	yes	—
285	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
286	C₂³	C₂×#10625325	1	1	yes	characteristic
287	C₂³	#13094057	1	2	yes	—
288	Q₈	—	376	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	94	8	no	—
306	C₂×C₄	—	188	2	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	—	188	2	no	—
311	C₂×C₄	—	188	2	no	—
312	C₂×C₄	—	188	4	no	—
313	C₂×C₄	—	188	4	no	—
314	C₂×C₄	—	188	4	no	—
315	C₂×C₄	—	188	4	no	—
316	C₂×C₄	—	188	4	no	—
317	C₂×C₄	—	188	4	no	—
318	C₂×C₄	#13093938	1	2	yes	—
319	C₂×C₄	#13093946	1	2	yes	—
320	C₂×C₄	#13093946	1	2	yes	—
321	C₂×C₄	#13093950	1	2	yes	—
322	C₂×C₄	#13093950	1	2	yes	—
323	C₂×C₄	#13093989	1	2	yes	—
324	C₂×C₄	#13094011	1	1	yes	characteristic
325	C₂×C₄	#13094022	1	1	yes	characteristic
326	C₈	—	8	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	#15689028	1	1	yes	characteristic
336	C₂²	C₂×#13093938	1	1	yes	characteristic
337	C₂²	#15689096	1	1	yes	characteristic
338	C₂²	C₂×#13093946	1	1	yes	characteristic
339	C₂²	#15689133	1	1	yes	characteristic
340	C₂²	C₂×#13093950	1	1	yes	characteristic
341	C₂²	C₂×#13093950	1	1	yes	characteristic
342	C₂²	#15689404	1	2	yes	—
343	C₂²	#15689409	1	2	yes	—
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	4	1	no	—
350	C₄	—	376	2	no	—
351	C₄	—	376	2	no	—
352	C₄	—	188	8	no	—
353	C₄	#15688858	1	4	yes	—
354	C₂	—	2	2	no	—
355	C₂	—	2	2	no	—
356	C₂	C₂×#15688858	1	1	yes	characteristic
357	C₂	#20241884	1	1	yes	characteristic
358	C₂	#20241901	1	1	yes	characteristic
359	C₂	#20241909	1	1	yes	characteristic
360	C₂	#20244339	1	1	yes	characteristic
361	C₂	#20244364	1	1	yes	characteristic
362	C₂	#20244372	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1797	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1797	D₄₇
hypocenter	C₄₇	#9909
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. C₂×#13094024	C₄	semidirect
10. #15689021	C₄	semidirect
11. #15688993	C₄	semidirect
39. C₄₇⋊#95	Q₈	semidirect
201. C₄₇	#9909	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20251740	2
C₄×C₄₇⋊#95	2
#20241268	2
C₄₇×#1797	2
#9909	47